7.4. Dot Product and Angle Between Two Vectors http://www.ck12.org
The dot product between the two vectors<− 1 , 1 >and< 4 , 4 >can be computed as:
(− 1 )( 4 )+ 1 ( 4 ) =− 4 + 4 = 0
The result of zero makes sense because these two vectors are perpendicular to each other.
Vocabulary
Thedot productis also known as inner productand scalar product. It is one of two kinds of products taken
between vectors. It produces a number that can be interpreted to tell how much one vector goes in the direction of
the other.
Guided Practice
- Show the distributive property holds under the dot product.
u·(v+w) =uv+uw - Find the dot product between the following vectors.
( 4 i− 2 j)·( 3 i− 8 j) - What is the angle betweenv=< 3 , 5 >andu=< 2 , 8 >?
Answers: - This proof will work with two dimensional vectors although the property does hold in general.
u=<u 1 ,u 2 >,v=<v 1 ,v 2 >,w=<w 1 ,w 2 >
u·(v+w) =u·(<v 1 ,v 2 >+<w 1 ,w 2 >)
=u·<v 1 +w 1 ,v 2 +w 2 >
=<u 1 ,u 2 >·<v 1 +w 1 ,v 2 +w 2 >
=u 1 (v 1 +w 1 )+u 2 (v 2 +w 2 )
=u 1 v 1 +u 1 w 1 +u 2 v 2 +u 2 w 2
=u 1 v 1 +u 2 v 2 +u 1 w 1 +u 2 w 2
=u·v+v·w- The standard unit vectors can be written as component vectors.
< 4 ,− 2 >·< 3 ,− 8 >= 12 +(− 2 )(− 8 ) = 12 + 16 = 28 - Use the angle between two vectors formula.
v=< 3 , 5 >andu=< 2 , 8 >